In this paper, we study two important extensions of the classical minimum cut problem, called {\em Connectivity Preserving Minimum Cut (CPMC)} problem and {\em Threshold Minimum Cut (TMC)} problem, which have important applications in large-scale DDoS attacks. In CPMC problem, a minimum cut is sought to separate a of source from a destination node and meanwhile preserve the connectivity between the source and its partner node(s). The CPMC problem also has important applications in many other areas such as emergency responding, image processing, pattern recognition, and medical sciences. In TMC problem, a minimum cut is sought to isolate a target node from a threshold number of partner nodes. TMC problem is an important special case of network inhibition problem and has important applications in network security. We show that the general CPMC problem cannot be approximated within $logn$ unless $NP=P$ has quasi-polynomial algorithms. We also show that a special case of two group CPMC problem in planar graphs can be solved in polynomial time. The corollary of this result is that the network diversion problem in planar graphs is in $P$, a previously open problem. We show that the threshold minimum node cut (TMNC) problem can be approximated within ratio $O(\sqrt{n})$ and the threshold minimum edge cut problem (TMEC) can be approximated within ratio $O(\log^2{n})$. \emph{We also answer another long standing open problem: the hardness of the network inhibition problem and network interdiction problem. We show that both of them cannot be approximated within any constant ratio. unless $NP \nsubseteq \cap_{δ>0} BPTIME(2^{n^δ})$.